The Picard-Lindelöf theorem states that a differential equation with $F:\mathbb{R}^n \rightarrow \mathbb{R}^n$ has a unique solution if $F$ is differentiable given some initial condition $x_0 \in \mathbb{R}^n$.
I am interested in the inverse problem. Given a solution $\gamma(x_0,t)$ for the maximal interval of existence, can we identify the ODE, i.e. $F$. Since this is impossible in general, I restrict to polynomial vector fields in $\mathbb{R}^n$ of degree $d$, i.e. $F \in \mathbb{R}[[x_1,...,x_n]]^d$. These functions are specified by $M={ \begin{pmatrix} N+d \\ d \end{pmatrix} }^{ N }$ real constants.
Is the following statement true?
Theorem: Given a trajectory $\gamma(x_0,t): \mathbb{R} \rightarrow \mathbb{R}^n$ such that the trajectory has at least $M$ different points $\{x_i\}_{i=1}^M \subset \mathbb{R}^m$. We also know the first derivative at the trajectory at these points $\dot{\gamma(x_i)} \quad \forall i =1,...,M$ which are also assumed to be different. We can reconstruct the polynomial vector field $F$ giving rise to the trajectory just by solving for the coefficients in the standard way.